The Quantum World of Ultra-Cold Atoms and Light, Book II: The Physics of Quantum-Optical Devices 9781783266159
This book, the second in our trilogy “The Quantum World of Ultra-Cold Atoms and Light”, aims to present the quantum-opti
276 20 6MB
English Pages 523 Year 2015
Table of contents :
Gardiner,C.W.,Zoller P.The Quantum World of Ultra-Cold Atoms and Light Book II The Physics of Quantum-Optical Devices(Gold Atom vol.2)(ICP,2015)(ISBN 9781783266159)(600dpi)(523p) ......Page 4
Copyright ......Page 5
Contents xiii ......Page 12
Preface vii ......Page 7
Acknowledgments xi ......Page 11
I Principles of Quantum Devices ......Page 25
1.1 Quantum Mechanics—from Paradox to Paradigm 3 ......Page 27
1.2.1 Quantum Optics—its Scope and its Aims 4 ......Page 28
1.2.3 Quantum Devices 5 ......Page 29
1.2.4 Quantum Information Theory 6 ......Page 30
1.3.1 The Quantum Optical System-Environment Paradigm 7 ......Page 31
1.3.3 Inputs and Outputs 8 ......Page 32
1.4 Quantum Control and Quantum Processing 9 ......Page 33
1.4.1 Implementing Quantum Control Using Trapped Ions 10 ......Page 34
1.4.2 Several Trapped Ions—the Quantum Processor 13 ......Page 37
1.5.1 Quantum Networks 14 ......Page 38
1.5.2 Cavity QED Systems 15 ......Page 39
1.6 Technologies for Quantum Processing 16 ......Page 40
1.7 Summary 17 ......Page 41
2.1 Representing and Processing Quantum Information 18 ......Page 42
2.1.1 Linear Unitary Evolution 19 ......Page 43
2.1.2 Quantum Parallelism and Projective Measurements 21 ......Page 45
2.2 The DiVincenzo Criteria for a Practical Quantum Computer 22 ......Page 46
2.3 Universal Sets of Quantum Gates 23 ......Page 47
2.3.2 Two-Qubit Gates 24 ......Page 48
2.3.3 Two Different Universal Sets of Gates 25 ......Page 49
2.4.1 Efficient Arithmetic Operations on a Quantum Computer 26 ......Page 50
2.4.2 Determination of Periodicities Using Quantum Measurements 27 ......Page 51
2.5.1 Efficient Quantum Simulation 29 ......Page 53
2.5.2 The Tasks for a Quantum Simulator 31 ......Page 55
3. Quantum Information 32 ......Page 56
3.1 Entanglement 33 ......Page 57
3.1.2 Bipartite Entangled States 34 ......Page 58
3.1.3 Bipartite Entanglement of Qubits 35 ......Page 59
3.1.4 The Schmidt Decomposition 37 ......Page 61
3.1.5 Multipartite Entanglement 39 ......Page 63
3.1.6 Decoherence, Mixed States and Entanglement 40 ......Page 64
3.2.1 The No-Cloning Theorem 42 ......Page 66
3.2.2 Teleportation 43 ......Page 67
II Coherent Optical Manipulation of Atoms ......Page 69
4.1 An Atom Driven by a Classical Driving Field 47 ......Page 71
4.1.1 Schrodinger Equation for a Multilevel Atom 48 ......Page 72
4.1.2 Non-Resonant Driving and the AC Stark Shift 49 ......Page 73
4.1.3 Two-Level Atoms and the AC Stark Shift 51 ......Page 75
5.1.1 General Pulse Shape 54 ......Page 78
5.2 The Rabi Problem for a Square Pulse 56 ......Page 80
5.2.1 The Pseudospin Formalism 57 ......Page 81
5.2.2 The Bloch Sphere 58 ......Page 82
5.2.3 The Bloch Equation 59 ......Page 83
5.3.1 Eigenvectors—the Dressed States 61 ......Page 85
5.3.3 Quantum State Engineering 62 ......Page 86
5.4.1 The Adiabatic Theorem and Berry’s Phase 66 ......Page 90
6.1 Three-Level Systems 69 ......Page 93
6.1.1 The A-Configuration 70 ......Page 94
6.2 Far-Detuned Raman Processes 72 ......Page 96
6.2.1 The Dark State Configuration 74 ......Page 98
6.2.2 Approximations in the Far-Detuned Case 75 ......Page 99
6.2.3 Effect of Spontaneous Emission 76 ......Page 100
6.3.1 Adiabatic Population Transfer along a Dark State—STIRAP 77 ......Page 101
6.3.2 Conditions for the Validity of the Adiabatic Transfer Procedure 78 ......Page 102
7. The Two-Level System Including Atomic Motion 80 ......Page 104
7.1.2 Basis States 81 ......Page 105
7.1.3 Equations of Motion 82 ......Page 106
7.2 Motion of an Atom in a Standing Light Wave 83 ......Page 107
7.2.2 Equations of Motion 84 ......Page 108
7.3 Momentum Transfer by Adiabatic Passage 87 ......Page 111
7.3.1 Solutions of the Equations of Motion 89 ......Page 113
8.1 The Alkali Atoms 90 ......Page 114
8.1.2 Electrostatic Interactions 91 ......Page 115
8.1.3 The Radial Wavefunction of the Valence Electron 92 ......Page 116
8.2.1 Spectroscopic Notation 94 ......Page 118
8.2.2 L-S Coupling of Angular Momenta 95 ......Page 119
8.3.1 Magnetic Moments 97 ......Page 121
8.3.3 Spin Orbit Interaction and Fine Structure 99 ......Page 123
8.3.4 Nuclear Spin and Hyperfine Structure 100 ......Page 124
8.3.5 The Zeeman Effect 103 ......Page 127
8.4 Interaction with Electromagnetic Radiation 104 ......Page 128
8.5.1 Electric Dipole Matrix Elements and Rabi Frequency 105 ......Page 129
8.5.2 Spontaneous Emission Rate for a Jg -> Je Transition 106 ......Page 130
8.5.3 Selection Rules, Polarization and Angular Distribution 108 ......Page 132
8.5.4 Total Transition Rate 109 ......Page 133
8.5.6 Fine Structure and Hyperfine Structure 110 ......Page 134
8.A Appendix: Spherical Tensors and the Wigner-Eckart Theorem 111......Page 135
III Atoms and Quantized Optical Fields ......Page 141
9. The Quantum Stochastic Schrodinger Equation 120 ......Page 144
9.1 The Quantum Optical System-Environment Model 121 ......Page 145
9.1.1 The Hamiltonian 122 ......Page 146
9.1.3 Examples of Hamiltonians and Coupling Operators 123 ......Page 147
9.2.1 The Schrodinger Equation in the Interaction Picture 124 ......Page 148
9.2.2 The Noise Operator 125 ......Page 149
9.2.3 The Born-Markov Approximation 126 ......Page 150
9.3 Formulation of Quantum Stochastic Calculus 128 ......Page 152
9.3.1 Coarse Graining in Time and the Quantum Ito Increments 129 ......Page 153
9.3.2 The Quantum Noise Hilbert Space 130 ......Page 154
9.3.3 Solution of the Schrodinger Equation for a Finite Time Interval 131 ......Page 155
9.3.4 Quantum Stochastic Schrodinger Equation in the Interaction Picture 132 ......Page 156
9.4.2 Quantum Stochastic Schrodinger Equation in the Schrodinger Picture 133 ......Page 157
9.4.3 The Central Role of the Quantum Stochastic Schrodinger Equation 134 ......Page 158
9.4.4 Solving the Quantum Stochastic Schrodinger Equation 135 ......Page 159
9.4.6 Coupling to Several Independent Light Fields 137 ......Page 161
9.5 Inclusion of a Coherent Input Field 138 ......Page 162
9.5.1 Quantum Stochastic Schrodinger Equation 139 ......Page 163
9.6.1 Finite Temperature Ito Increments 140 ......Page 164
9.6.2 Ensemble Formulation for Finite Temperature Fields 141 ......Page 165
9.6.3 Derivation of the Quantum Stochastic Differential Equation 142 ......Page 166
9.6.4 Evaluation of Thermal Averages 143 ......Page 167
9.6.5 Finite Temperature Evolution Operator 144 ......Page 168
10. The Master Equation 146 ......Page 170
10.1.1 Derivation of the Master Equation 147 ......Page 171
10.1.3 Non-Lindblad Master Equations 149 ......Page 173
10.2.2 Relation to the Full Evolution Operator 150 ......Page 174
10.2.3 Multitime Averages 151 ......Page 175
10.3 Applications of the Master Equation 152 ......Page 176
11. Inputs, Outputs and Quantum Langevin Equations 154 ......Page 178
11.1 The Quantum Langevin Equation 155 ......Page 179
11.1.1 The Quantum Langevin Equation 156 ......Page 180
11.2 The Quantum Langevin Equation in Terms of Outputs 157 ......Page 181
11.2.2 Time-Reversed Quantum Langevin Equation 158 ......Page 182
11.3 Correlation Functions of the Output 159 ......Page 183
11.3.1 Output Correlations Functions and System Correlation Functions 160 ......Page 184
11.4 Quantum Stochastic Differential Equations 161 ......Page 185
11.4.2 Alternative Form 162 ......Page 186
11.4.3 Correspondence with the Quantum Langevin Equation 163 ......Page 187
11.4.4 Phase Shift on Reflection From a Cavity 165 ......Page 189
11.5 Several Inputs and Outputs 166 ......Page 190
11.6 Fermionic Input-Output Theory 168 ......Page 192
12. Cascaded Quantum Systems 169 ......Page 193
12.1.1 Interpretation of the Quantum Langevin Equation 171 ......Page 195
12.2.1 Quantum Stochastic Differential Equation 172 ......Page 196
12.3.1 Driving a Quantum System with Light of Arbitrary Statistics 173 ......Page 197
12.3.2 Cascaded Quantum Networks 174 ......Page 198
13.1.1 System Hamiltonian 175 ......Page 199
13.1.2 The Master Equation 176 ......Page 200
13.2 Explicit Form and Solutions of the Optical Bloch Equations 177 ......Page 201
13.2.2 Stationary Solution in the Presence of a Coherent Driving Field 178 ......Page 202
13.2.4 Time-Dependent Solutions 179 ......Page 203
13.3 Dissipative Dynamics of the A-System 180 ......Page 204
13.3.1 Parameters for the Three-Level System 181 ......Page 205
13.3.2 System Hamiltonian 182 ......Page 206
13.3.3 Master Equation 183 ......Page 207
14.1 Coupling to Several Non-Independent Light Fields 185 ......Page 209
14.1.3 Evaluation of Damping Constants 186 ......Page 210
14.1.4 Evaluation of Lineshifts 188 ......Page 212
14.2.2 Quantum Stochastic Differential Equation 190 ......Page 214
14.3.1 Spontaneous Emission Rates and Dipole-Dipole Forces 191 ......Page 215
14.3.2 Special Cases 192 ......Page 216
14.3.3 Superradiance 194 ......Page 218
14.A Appendix: Spherical Bessel Functions 198 ......Page 222
IV Laser Cooling ......Page 223
15. Quantum Stochastic Equations for Laser Cooling of Atoms 201 ......Page 225
15.1.2 System Hamiltonian 202 ......Page 226
15.2 Quantum Stochastic Differential Equation Formalism 203 ......Page 227
15.2.3 Density of States 204 ......Page 228
15.2.5 Quantum Stochastic Schrodinger Equation 205 ......Page 229
15.3 Recoil Effects 206 ......Page 230
15.3.2 The Atom 207 ......Page 231
15.3.5 Quantum Stochastic Schrodinger Equation Including Atomic Motion 208 ......Page 232
15.4 The Laser Cooling Master Equation 210 ......Page 234
16.1 Wigner Representation of the Atomic Density Matrix 212 ......Page 236
16.1.2 Evolution Equation for the Wigner Distribution Function 213 ......Page 237
16.2 Doppler Cooling of a Two-Level System Using a Travelling Light Wave 214 ......Page 238
16.2.2 Evolution Operators and Stationary Solutions 215 ......Page 239
16.2.3 Projectors 219 ......Page 243
16.2.4 Adiabatic Elimination Procedure 222 ......Page 246
16.2.5 The Laser Cooling Fokker-Planck Equation 223 ......Page 247
16.2.6 Evaluation of the Diffusion Coefficient T)zz 224 ......Page 248
16.3 Summary 226 ......Page 250
17.1.1 Hamiltonian Terms 228 ......Page 252
17.1.2 Laser Cooling Terms 229 ......Page 253
17.2.1 Perturbative Expansion of the Master Equation 230 ......Page 254
17.2.2 Elimination of the Internal Degrees of Freedom 232 ......Page 256
17.3.1 The Harmonic Oscillator Trap 234 ......Page 258
17.3.2 Solutions of the Ion Trap Master Equation 236 ......Page 260
17.4.1 Doppler Cooling of a Trapped Ion 239 ......Page 263
17.4.2 Sideband Cooling 240 ......Page 264
V Continuous Measurement and Quantum Trajectories ......Page 265
18. Continuous Measurement 243 ......Page 267
18.1.1 The Normally Ordered Counting Formulae 244 ......Page 268
18.1.2 Measurement Operators for the Electromagnetic Field 245 ......Page 269
18.2 Photon Counting Formulae 246 ......Page 270
18.2.1 Photon Counting Statistics 248 ......Page 272
18.2.2 Classical and Non-Classical Light 250 ......Page 274
18.3 Quantum Operations 251 ......Page 275
18.3.1 Definition of a Quantum Operation 252 ......Page 276
18.3.2 Quantum Operations on the Wavefunction 254 ......Page 278
18.3.3 Measurement Operators for the Electromagnetic Field 255 ......Page 279
18.4 Photon Counting Using the Quantum Stochastic Schrodinger Equation 256 ......Page 280
18.4.2 Formulation of Photon Counting Using Quantum Operations 257 ......Page 281
18.4.3 Quantum Operations Induced on the System 259 ......Page 283
18.4.4 Probability Density of the First Detection Time 260 ......Page 284
18.4.5 Resolution in Terms of Photon Counts 261 ......Page 285
18.5.1 Resolution of the System Density Operator 262 ......Page 286
18.5.2 Comparison with the Classical Poisson Process 263 ......Page 287
18.6.1 Photon Counting Statistics 265 ......Page 289
18.6.2 Density Operator after ft Counts 266 ......Page 290
19. Quantum Trajectories 267 ......Page 291
19.1.1 Stochastic Interpretation 268 ......Page 292
19.1.3 Probabilities 269 ......Page 293
19.3 Applications 270 ......Page 294
19.3.1 Spontaneous Emission from a Two-Level Atom 271 ......Page 295
19.3.2 The Driven Two-Level System 273 ......Page 297
19.3.3 The Damped Cavity Mode 276 ......Page 300
19.4 Quantum Jumps in Three-Level Systems 279 ......Page 303
19.4.1 Theoretical Description 280 ......Page 304
19.4.2 Photon Detections in Terms of the Delay Function 281 ......Page 305
VI Phase-Sensitive Quantum Optics ......Page 307
20. Homodyne Measurement 285 ......Page 309
20.1.1 General Formulae 286 ......Page 310
20.1.2 Coherent Signal Detection 288 ......Page 312
20.1.3 Balanced Homodyne/Heterodyne Detection 290 ......Page 314
20.A.1 Ideal Homodyne Measurement 292 ......Page 316
20.A.4 Measurement Operators for Eigenvalues with a Continuous Range 293 ......Page 317
20.A.5 Formulation of the Quantum Stochastic Schrodinger Equation 294 ......Page 318
20.A.7 Continuous Measurement of the Quadrature Phase Components 295 ......Page 319
21. Squeezing, Quantum Correlations and Quantum Amplifiers 301 ......Page 325
21.1.1 Heisenberg's Uncertainty Principle 302 ......Page 326
21.1.3 Defining Squeezing 303 ......Page 327
21.1.4 Squeezed States of the Harmonic Oscillator 304 ......Page 328
21.1.5 Definition of an Ideal Squeezed State 306 ......Page 330
21.2 Production and Measurement of Squeezed Light 308 ......Page 332
21.2.1 The Degenerate Parametric Amplifier 309 ......Page 333
21.2.2 Quantum Langevin Equation 310 ......Page 334
21.2.3 Squeezing Produced 312 ......Page 336
21.2.4 Input-Output View of Squeezing and the Ideal Phase-Sensitive Amplifier 314 ......Page 338
21.3 Two-Mode Squeezing and Correlated Quanta 315 ......Page 339
21.4 Quantum Limited Phase-Insensitive Amplifiers 316 ......Page 340
21.4.1 Identical Couplings to the Input and Output 317 ......Page 341
21.4.2 Input and Output Coupled to Only One Mode 319 ......Page 343
21.5.1 Fundamental Limits on Added Quantum Noise—the Caves Amplifier Noise Bound 321 ......Page 345
21.5.2 Attenuators and Beam Splitters 322 ......Page 346
VII Quantum Processing with Atoms, Photons and Phonons ......Page 349
22.1 The Jaynes-Cummings Model 327 ......Page 351
22.1.1 Implementations of Cavity Quantum Electrodynamics 328 ......Page 352
22.1.2 Details of the Jaynes-Cummings Hamiltonian 329 ......Page 353
22.1.3 Energy Levels and Eigenstates 330 ......Page 354
22.1.4 Eigenvalues and Eigenvectors—the Dressed States 331 ......Page 355
22.1.5 The Dressed Hamiltonian in the Far-Detuned Limit 333 ......Page 357
22.1.6 Rabi Oscillations in the Jaynes-Cummings Model 334 ......Page 358
22.2.1 Master Equation 336 ......Page 360
22.2.2 The Strong Coupling Condition 337 ......Page 361
23.1 The Trapped Ion Hamiltonian 339 ......Page 363
23.1.1 Components of the Trapped Ion Hamiltonian 340 ......Page 364
23.1.4 Manipulation of the Quantum State of the Centre of Mass 341 ......Page 365
23.1.6 Total Hamiltonian in the Rotating Frame 342 ......Page 366
23.1.7 Energy Spectrum and Eigenstates of the Bare Hamiltonian .343 ......Page 367
23.2 Laser-Induced Couplings in the Lamb-Dicke Regime 344 ......Page 368
23.2.1 Approximate Forms in the Lamb-Dicke Regime 345 ......Page 369
23.2.2 Effective Hamiltonians Arising from Laser Induced Couplings 346 ......Page 370
23.3.2 Conversion of Electronic Superpositions to Motional Superpositions 347 ......Page 371
23.3.3 Generation of an Arbitrary Superposition of Motional States 348 ......Page 372
23.A.1 Classical Equations of Motion 350 ......Page 374
23.A.2 Quantum Theory of Ion Traps 354 ......Page 378
24. The Ion Trap Quantum Computer 358 ......Page 382
24.1.1 Two Ions in a Linear Trap 359 ......Page 383
24.1.2 N Atoms in a Linear Trap 361 ......Page 385
24.2 Implementation of a Two-Qubit Quantum Gate 362 ......Page 386
24.2.2 Manipulation of Qubits 363 ......Page 387
24.2.3 Basic Gate Operations 364 ......Page 388
24.2.4 The Controlled Phase Gate 366 ......Page 390
24.3 Molmer-Sorensen Gate 368 ......Page 392
24.3.1 Gate Hamiltonian 369 ......Page 393
24.3.2 Transition Paths 370 ......Page 394
24.3.4 Use as a Two-Qubit Gate 372 ......Page 396
24.3.5 Creation of GHZ-Like States 373 ......Page 397
24.3.6 Creation of Many-Body Entanglement 374 ......Page 398
24.4.1 Geometric Phase in a Harmonic Oscillator 375 ......Page 399
24.4.2 Phase of Two Ions 377 ......Page 401
24.5 The Ion Trap Quantum Computer in Practice 379 ......Page 403
24.5.1 Quantum Computing Using Multiple Ion Traps on a Chip 380 ......Page 404
VIII Circuit Quantum Electrodynamics ......Page 405
25.1 The LC Oscillator 383 ......Page 407
25.1.1 Lagrangian and Hamiltonian 384 ......Page 408
25.1.2 Quantization of the Oscillator 385 ......Page 409
25.2.1 Transmission Line Wave Equation 388 ......Page 412
25.2.2 The Flux Potential and the Lagrangian Formulation 389 ......Page 413
25.2.3 Boundary Conditions 390 ......Page 414
25.3.1 Parallel Coupling of the LC Oscillator to a Transmission Line 391 ......Page 415
25.3.2 Series Coupling of the LC Oscillator to a Transmission Line 393 ......Page 417
25.3.3 An Oscillator Embedded in a Transmission Line 394 ......Page 418
25.4.1 Voltage Coupling 396 ......Page 420
25.5.1 The Finite Transmission Line 398 ......Page 422
25.5.2 Coupling to an LC Circuit 399 ......Page 423
25.A.3 Time-Dependent Source of Fixed Shape 400 ......Page 424
26.1 The Josephson junction 402 ......Page 426
26.1.1 Analysis of the Hamiltonian 403 ......Page 427
26.1.3 Operator Form of the Hamiltonian 405 ......Page 429
26.1.4 Equations of Motion fora losephson lunction 406 ......Page 430
26.1.5 The losephson Junction as a Non-Linear Inductor 407 ......Page 431
26.2.1 Josephson Oscillations 408 ......Page 432
26.2.2 The Open Circuit Configuration 410 ......Page 434
26.3 Qubit Architectures 411 ......Page 435
26.3.1 The Transmon Qubit 412 ......Page 436
26.3.2 Formulation as Cavity QED 413 ......Page 437
26.3.3 Measurement of the Output 416 ......Page 440
26.4 Josephson Junction Amplification 417 ......Page 441
26.4.1 The Josephson Ring Modulator 418 ......Page 442
IX Interfacing Quantum Networks ......Page 445
27.1 The Cavity QED Quantum Memory 423 ......Page 447
27.1.2 Adiabatic Elimination of the Excited State 424 ......Page 448
27.1.3 Interaction with Input and Output Fields 425 ......Page 449
27.1.5 A Programmable Single Photon Source 427 ......Page 451
27.2.2 The Cavity QED Model of Quantum Information Transmission 429 ......Page 453
27.2.3 Achieving Ideal Quantum Transmission 430 ......Page 454
27.2.4 Use of the Quantum Trajectory Picture 431 ......Page 455
28. The Dark-State Ensemble Quantum Memory 434 ......Page 458
28.1.1 Hamiltonian for N Three-Level Atoms 435 ......Page 459
28.1.2 The Family of Dark States 437 ......Page 461
28.2.1 The Harmonic Approximation to an Ensemble of Atoms 439 ......Page 463
28.2.2 Equations of Motion in the Harmonic Oscillator Approximation 441 ......Page 465
28.3 Quantum State Transfer and Quantum Memory 442 ......Page 466
28.3.1 The A-I Configuration 443 ......Page 467
28.3.2 The Information Transfer Process 445 ......Page 469
28.3.4 Transferring the Stored Input to the Output 448 ......Page 472
28.3.5 The A-II Configuration 450 ......Page 474
29.1 Light Propagation in an Atomic Vapour—Semiclassical Theory 453 ......Page 477
29.1.1 Electric Polarization and Susceptibility 454 ......Page 478
29.1.2 Perturbative Regime in the Probe Field—Susceptibility 456 ......Page 480
29.2 Propagation in One Dimension 457 ......Page 481
29.2.1 The Wave Equation 458 ......Page 482
29.2.2 Absorptive and Dispersive Behaviour 459 ......Page 483
29.2.3 The Transparency Window 461 ......Page 485
29.2.4 Slow Light 462 ......Page 486
29.3.1 One-Dimensional Electromagnetic Field Operators 464 ......Page 488
29.3.2 Spatially Dependent Collective Atomic Operators 465 ......Page 489
29.3.3 Hamiltonian and Equations of Motion 466 ......Page 490
29.3.4 Solutions of the Equations of Motion 467 ......Page 491
29.4.1 Interpretation as Dark-State Polaritons 468 ......Page 492
29.4.2 Stopping and Re-Accelerating Photon Wavepackets 469 ......Page 493
References 470 ......Page 494
Author Index 483 ......Page 507
Subject Index 485 ......Page 509
cover......Page 1
Gardiner,C.W.,Zoller P.The Quantum World of Ultra-Cold Atoms and Light Book II The Physics of Quantum-Optical Devices(Gold Atom vol.2)(ICP,2015)(ISBN 9781783266159)(600dpi)(523p) ......Page 4
Copyright ......Page 5
Contents xiii ......Page 12
Preface vii ......Page 7
Acknowledgments xi ......Page 11
I Principles of Quantum Devices ......Page 25
1.1 Quantum Mechanics—from Paradox to Paradigm 3 ......Page 27
1.2.1 Quantum Optics—its Scope and its Aims 4 ......Page 28
1.2.3 Quantum Devices 5 ......Page 29
1.2.4 Quantum Information Theory 6 ......Page 30
1.3.1 The Quantum Optical System-Environment Paradigm 7 ......Page 31
1.3.3 Inputs and Outputs 8 ......Page 32
1.4 Quantum Control and Quantum Processing 9 ......Page 33
1.4.1 Implementing Quantum Control Using Trapped Ions 10 ......Page 34
1.4.2 Several Trapped Ions—the Quantum Processor 13 ......Page 37
1.5.1 Quantum Networks 14 ......Page 38
1.5.2 Cavity QED Systems 15 ......Page 39
1.6 Technologies for Quantum Processing 16 ......Page 40
1.7 Summary 17 ......Page 41
2.1 Representing and Processing Quantum Information 18 ......Page 42
2.1.1 Linear Unitary Evolution 19 ......Page 43
2.1.2 Quantum Parallelism and Projective Measurements 21 ......Page 45
2.2 The DiVincenzo Criteria for a Practical Quantum Computer 22 ......Page 46
2.3 Universal Sets of Quantum Gates 23 ......Page 47
2.3.2 Two-Qubit Gates 24 ......Page 48
2.3.3 Two Different Universal Sets of Gates 25 ......Page 49
2.4.1 Efficient Arithmetic Operations on a Quantum Computer 26 ......Page 50
2.4.2 Determination of Periodicities Using Quantum Measurements 27 ......Page 51
2.5.1 Efficient Quantum Simulation 29 ......Page 53
2.5.2 The Tasks for a Quantum Simulator 31 ......Page 55
3. Quantum Information 32 ......Page 56
3.1 Entanglement 33 ......Page 57
3.1.2 Bipartite Entangled States 34 ......Page 58
3.1.3 Bipartite Entanglement of Qubits 35 ......Page 59
3.1.4 The Schmidt Decomposition 37 ......Page 61
3.1.5 Multipartite Entanglement 39 ......Page 63
3.1.6 Decoherence, Mixed States and Entanglement 40 ......Page 64
3.2.1 The No-Cloning Theorem 42 ......Page 66
3.2.2 Teleportation 43 ......Page 67
II Coherent Optical Manipulation of Atoms ......Page 69
4.1 An Atom Driven by a Classical Driving Field 47 ......Page 71
4.1.1 Schrodinger Equation for a Multilevel Atom 48 ......Page 72
4.1.2 Non-Resonant Driving and the AC Stark Shift 49 ......Page 73
4.1.3 Two-Level Atoms and the AC Stark Shift 51 ......Page 75
5.1.1 General Pulse Shape 54 ......Page 78
5.2 The Rabi Problem for a Square Pulse 56 ......Page 80
5.2.1 The Pseudospin Formalism 57 ......Page 81
5.2.2 The Bloch Sphere 58 ......Page 82
5.2.3 The Bloch Equation 59 ......Page 83
5.3.1 Eigenvectors—the Dressed States 61 ......Page 85
5.3.3 Quantum State Engineering 62 ......Page 86
5.4.1 The Adiabatic Theorem and Berry’s Phase 66 ......Page 90
6.1 Three-Level Systems 69 ......Page 93
6.1.1 The A-Configuration 70 ......Page 94
6.2 Far-Detuned Raman Processes 72 ......Page 96
6.2.1 The Dark State Configuration 74 ......Page 98
6.2.2 Approximations in the Far-Detuned Case 75 ......Page 99
6.2.3 Effect of Spontaneous Emission 76 ......Page 100
6.3.1 Adiabatic Population Transfer along a Dark State—STIRAP 77 ......Page 101
6.3.2 Conditions for the Validity of the Adiabatic Transfer Procedure 78 ......Page 102
7. The Two-Level System Including Atomic Motion 80 ......Page 104
7.1.2 Basis States 81 ......Page 105
7.1.3 Equations of Motion 82 ......Page 106
7.2 Motion of an Atom in a Standing Light Wave 83 ......Page 107
7.2.2 Equations of Motion 84 ......Page 108
7.3 Momentum Transfer by Adiabatic Passage 87 ......Page 111
7.3.1 Solutions of the Equations of Motion 89 ......Page 113
8.1 The Alkali Atoms 90 ......Page 114
8.1.2 Electrostatic Interactions 91 ......Page 115
8.1.3 The Radial Wavefunction of the Valence Electron 92 ......Page 116
8.2.1 Spectroscopic Notation 94 ......Page 118
8.2.2 L-S Coupling of Angular Momenta 95 ......Page 119
8.3.1 Magnetic Moments 97 ......Page 121
8.3.3 Spin Orbit Interaction and Fine Structure 99 ......Page 123
8.3.4 Nuclear Spin and Hyperfine Structure 100 ......Page 124
8.3.5 The Zeeman Effect 103 ......Page 127
8.4 Interaction with Electromagnetic Radiation 104 ......Page 128
8.5.1 Electric Dipole Matrix Elements and Rabi Frequency 105 ......Page 129
8.5.2 Spontaneous Emission Rate for a Jg -> Je Transition 106 ......Page 130
8.5.3 Selection Rules, Polarization and Angular Distribution 108 ......Page 132
8.5.4 Total Transition Rate 109 ......Page 133
8.5.6 Fine Structure and Hyperfine Structure 110 ......Page 134
8.A Appendix: Spherical Tensors and the Wigner-Eckart Theorem 111......Page 135
III Atoms and Quantized Optical Fields ......Page 141
9. The Quantum Stochastic Schrodinger Equation 120 ......Page 144
9.1 The Quantum Optical System-Environment Model 121 ......Page 145
9.1.1 The Hamiltonian 122 ......Page 146
9.1.3 Examples of Hamiltonians and Coupling Operators 123 ......Page 147
9.2.1 The Schrodinger Equation in the Interaction Picture 124 ......Page 148
9.2.2 The Noise Operator 125 ......Page 149
9.2.3 The Born-Markov Approximation 126 ......Page 150
9.3 Formulation of Quantum Stochastic Calculus 128 ......Page 152
9.3.1 Coarse Graining in Time and the Quantum Ito Increments 129 ......Page 153
9.3.2 The Quantum Noise Hilbert Space 130 ......Page 154
9.3.3 Solution of the Schrodinger Equation for a Finite Time Interval 131 ......Page 155
9.3.4 Quantum Stochastic Schrodinger Equation in the Interaction Picture 132 ......Page 156
9.4.2 Quantum Stochastic Schrodinger Equation in the Schrodinger Picture 133 ......Page 157
9.4.3 The Central Role of the Quantum Stochastic Schrodinger Equation 134 ......Page 158
9.4.4 Solving the Quantum Stochastic Schrodinger Equation 135 ......Page 159
9.4.6 Coupling to Several Independent Light Fields 137 ......Page 161
9.5 Inclusion of a Coherent Input Field 138 ......Page 162
9.5.1 Quantum Stochastic Schrodinger Equation 139 ......Page 163
9.6.1 Finite Temperature Ito Increments 140 ......Page 164
9.6.2 Ensemble Formulation for Finite Temperature Fields 141 ......Page 165
9.6.3 Derivation of the Quantum Stochastic Differential Equation 142 ......Page 166
9.6.4 Evaluation of Thermal Averages 143 ......Page 167
9.6.5 Finite Temperature Evolution Operator 144 ......Page 168
10. The Master Equation 146 ......Page 170
10.1.1 Derivation of the Master Equation 147 ......Page 171
10.1.3 Non-Lindblad Master Equations 149 ......Page 173
10.2.2 Relation to the Full Evolution Operator 150 ......Page 174
10.2.3 Multitime Averages 151 ......Page 175
10.3 Applications of the Master Equation 152 ......Page 176
11. Inputs, Outputs and Quantum Langevin Equations 154 ......Page 178
11.1 The Quantum Langevin Equation 155 ......Page 179
11.1.1 The Quantum Langevin Equation 156 ......Page 180
11.2 The Quantum Langevin Equation in Terms of Outputs 157 ......Page 181
11.2.2 Time-Reversed Quantum Langevin Equation 158 ......Page 182
11.3 Correlation Functions of the Output 159 ......Page 183
11.3.1 Output Correlations Functions and System Correlation Functions 160 ......Page 184
11.4 Quantum Stochastic Differential Equations 161 ......Page 185
11.4.2 Alternative Form 162 ......Page 186
11.4.3 Correspondence with the Quantum Langevin Equation 163 ......Page 187
11.4.4 Phase Shift on Reflection From a Cavity 165 ......Page 189
11.5 Several Inputs and Outputs 166 ......Page 190
11.6 Fermionic Input-Output Theory 168 ......Page 192
12. Cascaded Quantum Systems 169 ......Page 193
12.1.1 Interpretation of the Quantum Langevin Equation 171 ......Page 195
12.2.1 Quantum Stochastic Differential Equation 172 ......Page 196
12.3.1 Driving a Quantum System with Light of Arbitrary Statistics 173 ......Page 197
12.3.2 Cascaded Quantum Networks 174 ......Page 198
13.1.1 System Hamiltonian 175 ......Page 199
13.1.2 The Master Equation 176 ......Page 200
13.2 Explicit Form and Solutions of the Optical Bloch Equations 177 ......Page 201
13.2.2 Stationary Solution in the Presence of a Coherent Driving Field 178 ......Page 202
13.2.4 Time-Dependent Solutions 179 ......Page 203
13.3 Dissipative Dynamics of the A-System 180 ......Page 204
13.3.1 Parameters for the Three-Level System 181 ......Page 205
13.3.2 System Hamiltonian 182 ......Page 206
13.3.3 Master Equation 183 ......Page 207
14.1 Coupling to Several Non-Independent Light Fields 185 ......Page 209
14.1.3 Evaluation of Damping Constants 186 ......Page 210
14.1.4 Evaluation of Lineshifts 188 ......Page 212
14.2.2 Quantum Stochastic Differential Equation 190 ......Page 214
14.3.1 Spontaneous Emission Rates and Dipole-Dipole Forces 191 ......Page 215
14.3.2 Special Cases 192 ......Page 216
14.3.3 Superradiance 194 ......Page 218
14.A Appendix: Spherical Bessel Functions 198 ......Page 222
IV Laser Cooling ......Page 223
15. Quantum Stochastic Equations for Laser Cooling of Atoms 201 ......Page 225
15.1.2 System Hamiltonian 202 ......Page 226
15.2 Quantum Stochastic Differential Equation Formalism 203 ......Page 227
15.2.3 Density of States 204 ......Page 228
15.2.5 Quantum Stochastic Schrodinger Equation 205 ......Page 229
15.3 Recoil Effects 206 ......Page 230
15.3.2 The Atom 207 ......Page 231
15.3.5 Quantum Stochastic Schrodinger Equation Including Atomic Motion 208 ......Page 232
15.4 The Laser Cooling Master Equation 210 ......Page 234
16.1 Wigner Representation of the Atomic Density Matrix 212 ......Page 236
16.1.2 Evolution Equation for the Wigner Distribution Function 213 ......Page 237
16.2 Doppler Cooling of a Two-Level System Using a Travelling Light Wave 214 ......Page 238
16.2.2 Evolution Operators and Stationary Solutions 215 ......Page 239
16.2.3 Projectors 219 ......Page 243
16.2.4 Adiabatic Elimination Procedure 222 ......Page 246
16.2.5 The Laser Cooling Fokker-Planck Equation 223 ......Page 247
16.2.6 Evaluation of the Diffusion Coefficient T)zz 224 ......Page 248
16.3 Summary 226 ......Page 250
17.1.1 Hamiltonian Terms 228 ......Page 252
17.1.2 Laser Cooling Terms 229 ......Page 253
17.2.1 Perturbative Expansion of the Master Equation 230 ......Page 254
17.2.2 Elimination of the Internal Degrees of Freedom 232 ......Page 256
17.3.1 The Harmonic Oscillator Trap 234 ......Page 258
17.3.2 Solutions of the Ion Trap Master Equation 236 ......Page 260
17.4.1 Doppler Cooling of a Trapped Ion 239 ......Page 263
17.4.2 Sideband Cooling 240 ......Page 264
V Continuous Measurement and Quantum Trajectories ......Page 265
18. Continuous Measurement 243 ......Page 267
18.1.1 The Normally Ordered Counting Formulae 244 ......Page 268
18.1.2 Measurement Operators for the Electromagnetic Field 245 ......Page 269
18.2 Photon Counting Formulae 246 ......Page 270
18.2.1 Photon Counting Statistics 248 ......Page 272
18.2.2 Classical and Non-Classical Light 250 ......Page 274
18.3 Quantum Operations 251 ......Page 275
18.3.1 Definition of a Quantum Operation 252 ......Page 276
18.3.2 Quantum Operations on the Wavefunction 254 ......Page 278
18.3.3 Measurement Operators for the Electromagnetic Field 255 ......Page 279
18.4 Photon Counting Using the Quantum Stochastic Schrodinger Equation 256 ......Page 280
18.4.2 Formulation of Photon Counting Using Quantum Operations 257 ......Page 281
18.4.3 Quantum Operations Induced on the System 259 ......Page 283
18.4.4 Probability Density of the First Detection Time 260 ......Page 284
18.4.5 Resolution in Terms of Photon Counts 261 ......Page 285
18.5.1 Resolution of the System Density Operator 262 ......Page 286
18.5.2 Comparison with the Classical Poisson Process 263 ......Page 287
18.6.1 Photon Counting Statistics 265 ......Page 289
18.6.2 Density Operator after ft Counts 266 ......Page 290
19. Quantum Trajectories 267 ......Page 291
19.1.1 Stochastic Interpretation 268 ......Page 292
19.1.3 Probabilities 269 ......Page 293
19.3 Applications 270 ......Page 294
19.3.1 Spontaneous Emission from a Two-Level Atom 271 ......Page 295
19.3.2 The Driven Two-Level System 273 ......Page 297
19.3.3 The Damped Cavity Mode 276 ......Page 300
19.4 Quantum Jumps in Three-Level Systems 279 ......Page 303
19.4.1 Theoretical Description 280 ......Page 304
19.4.2 Photon Detections in Terms of the Delay Function 281 ......Page 305
VI Phase-Sensitive Quantum Optics ......Page 307
20. Homodyne Measurement 285 ......Page 309
20.1.1 General Formulae 286 ......Page 310
20.1.2 Coherent Signal Detection 288 ......Page 312
20.1.3 Balanced Homodyne/Heterodyne Detection 290 ......Page 314
20.A.1 Ideal Homodyne Measurement 292 ......Page 316
20.A.4 Measurement Operators for Eigenvalues with a Continuous Range 293 ......Page 317
20.A.5 Formulation of the Quantum Stochastic Schrodinger Equation 294 ......Page 318
20.A.7 Continuous Measurement of the Quadrature Phase Components 295 ......Page 319
21. Squeezing, Quantum Correlations and Quantum Amplifiers 301 ......Page 325
21.1.1 Heisenberg's Uncertainty Principle 302 ......Page 326
21.1.3 Defining Squeezing 303 ......Page 327
21.1.4 Squeezed States of the Harmonic Oscillator 304 ......Page 328
21.1.5 Definition of an Ideal Squeezed State 306 ......Page 330
21.2 Production and Measurement of Squeezed Light 308 ......Page 332
21.2.1 The Degenerate Parametric Amplifier 309 ......Page 333
21.2.2 Quantum Langevin Equation 310 ......Page 334
21.2.3 Squeezing Produced 312 ......Page 336
21.2.4 Input-Output View of Squeezing and the Ideal Phase-Sensitive Amplifier 314 ......Page 338
21.3 Two-Mode Squeezing and Correlated Quanta 315 ......Page 339
21.4 Quantum Limited Phase-Insensitive Amplifiers 316 ......Page 340
21.4.1 Identical Couplings to the Input and Output 317 ......Page 341
21.4.2 Input and Output Coupled to Only One Mode 319 ......Page 343
21.5.1 Fundamental Limits on Added Quantum Noise—the Caves Amplifier Noise Bound 321 ......Page 345
21.5.2 Attenuators and Beam Splitters 322 ......Page 346
VII Quantum Processing with Atoms, Photons and Phonons ......Page 349
22.1 The Jaynes-Cummings Model 327 ......Page 351
22.1.1 Implementations of Cavity Quantum Electrodynamics 328 ......Page 352
22.1.2 Details of the Jaynes-Cummings Hamiltonian 329 ......Page 353
22.1.3 Energy Levels and Eigenstates 330 ......Page 354
22.1.4 Eigenvalues and Eigenvectors—the Dressed States 331 ......Page 355
22.1.5 The Dressed Hamiltonian in the Far-Detuned Limit 333 ......Page 357
22.1.6 Rabi Oscillations in the Jaynes-Cummings Model 334 ......Page 358
22.2.1 Master Equation 336 ......Page 360
22.2.2 The Strong Coupling Condition 337 ......Page 361
23.1 The Trapped Ion Hamiltonian 339 ......Page 363
23.1.1 Components of the Trapped Ion Hamiltonian 340 ......Page 364
23.1.4 Manipulation of the Quantum State of the Centre of Mass 341 ......Page 365
23.1.6 Total Hamiltonian in the Rotating Frame 342 ......Page 366
23.1.7 Energy Spectrum and Eigenstates of the Bare Hamiltonian .343 ......Page 367
23.2 Laser-Induced Couplings in the Lamb-Dicke Regime 344 ......Page 368
23.2.1 Approximate Forms in the Lamb-Dicke Regime 345 ......Page 369
23.2.2 Effective Hamiltonians Arising from Laser Induced Couplings 346 ......Page 370
23.3.2 Conversion of Electronic Superpositions to Motional Superpositions 347 ......Page 371
23.3.3 Generation of an Arbitrary Superposition of Motional States 348 ......Page 372
23.A.1 Classical Equations of Motion 350 ......Page 374
23.A.2 Quantum Theory of Ion Traps 354 ......Page 378
24. The Ion Trap Quantum Computer 358 ......Page 382
24.1.1 Two Ions in a Linear Trap 359 ......Page 383
24.1.2 N Atoms in a Linear Trap 361 ......Page 385
24.2 Implementation of a Two-Qubit Quantum Gate 362 ......Page 386
24.2.2 Manipulation of Qubits 363 ......Page 387
24.2.3 Basic Gate Operations 364 ......Page 388
24.2.4 The Controlled Phase Gate 366 ......Page 390
24.3 Molmer-Sorensen Gate 368 ......Page 392
24.3.1 Gate Hamiltonian 369 ......Page 393
24.3.2 Transition Paths 370 ......Page 394
24.3.4 Use as a Two-Qubit Gate 372 ......Page 396
24.3.5 Creation of GHZ-Like States 373 ......Page 397
24.3.6 Creation of Many-Body Entanglement 374 ......Page 398
24.4.1 Geometric Phase in a Harmonic Oscillator 375 ......Page 399
24.4.2 Phase of Two Ions 377 ......Page 401
24.5 The Ion Trap Quantum Computer in Practice 379 ......Page 403
24.5.1 Quantum Computing Using Multiple Ion Traps on a Chip 380 ......Page 404
VIII Circuit Quantum Electrodynamics ......Page 405
25.1 The LC Oscillator 383 ......Page 407
25.1.1 Lagrangian and Hamiltonian 384 ......Page 408
25.1.2 Quantization of the Oscillator 385 ......Page 409
25.2.1 Transmission Line Wave Equation 388 ......Page 412
25.2.2 The Flux Potential and the Lagrangian Formulation 389 ......Page 413
25.2.3 Boundary Conditions 390 ......Page 414
25.3.1 Parallel Coupling of the LC Oscillator to a Transmission Line 391 ......Page 415
25.3.2 Series Coupling of the LC Oscillator to a Transmission Line 393 ......Page 417
25.3.3 An Oscillator Embedded in a Transmission Line 394 ......Page 418
25.4.1 Voltage Coupling 396 ......Page 420
25.5.1 The Finite Transmission Line 398 ......Page 422
25.5.2 Coupling to an LC Circuit 399 ......Page 423
25.A.3 Time-Dependent Source of Fixed Shape 400 ......Page 424
26.1 The Josephson junction 402 ......Page 426
26.1.1 Analysis of the Hamiltonian 403 ......Page 427
26.1.3 Operator Form of the Hamiltonian 405 ......Page 429
26.1.4 Equations of Motion fora losephson lunction 406 ......Page 430
26.1.5 The losephson Junction as a Non-Linear Inductor 407 ......Page 431
26.2.1 Josephson Oscillations 408 ......Page 432
26.2.2 The Open Circuit Configuration 410 ......Page 434
26.3 Qubit Architectures 411 ......Page 435
26.3.1 The Transmon Qubit 412 ......Page 436
26.3.2 Formulation as Cavity QED 413 ......Page 437
26.3.3 Measurement of the Output 416 ......Page 440
26.4 Josephson Junction Amplification 417 ......Page 441
26.4.1 The Josephson Ring Modulator 418 ......Page 442
IX Interfacing Quantum Networks ......Page 445
27.1 The Cavity QED Quantum Memory 423 ......Page 447
27.1.2 Adiabatic Elimination of the Excited State 424 ......Page 448
27.1.3 Interaction with Input and Output Fields 425 ......Page 449
27.1.5 A Programmable Single Photon Source 427 ......Page 451
27.2.2 The Cavity QED Model of Quantum Information Transmission 429 ......Page 453
27.2.3 Achieving Ideal Quantum Transmission 430 ......Page 454
27.2.4 Use of the Quantum Trajectory Picture 431 ......Page 455
28. The Dark-State Ensemble Quantum Memory 434 ......Page 458
28.1.1 Hamiltonian for N Three-Level Atoms 435 ......Page 459
28.1.2 The Family of Dark States 437 ......Page 461
28.2.1 The Harmonic Approximation to an Ensemble of Atoms 439 ......Page 463
28.2.2 Equations of Motion in the Harmonic Oscillator Approximation 441 ......Page 465
28.3 Quantum State Transfer and Quantum Memory 442 ......Page 466
28.3.1 The A-I Configuration 443 ......Page 467
28.3.2 The Information Transfer Process 445 ......Page 469
28.3.4 Transferring the Stored Input to the Output 448 ......Page 472
28.3.5 The A-II Configuration 450 ......Page 474
29.1 Light Propagation in an Atomic Vapour—Semiclassical Theory 453 ......Page 477
29.1.1 Electric Polarization and Susceptibility 454 ......Page 478
29.1.2 Perturbative Regime in the Probe Field—Susceptibility 456 ......Page 480
29.2 Propagation in One Dimension 457 ......Page 481
29.2.1 The Wave Equation 458 ......Page 482
29.2.2 Absorptive and Dispersive Behaviour 459 ......Page 483
29.2.3 The Transparency Window 461 ......Page 485
29.2.4 Slow Light 462 ......Page 486
29.3.1 One-Dimensional Electromagnetic Field Operators 464 ......Page 488
29.3.2 Spatially Dependent Collective Atomic Operators 465 ......Page 489
29.3.3 Hamiltonian and Equations of Motion 466 ......Page 490
29.3.4 Solutions of the Equations of Motion 467 ......Page 491
29.4.1 Interpretation as Dark-State Polaritons 468 ......Page 492
29.4.2 Stopping and Re-Accelerating Photon Wavepackets 469 ......Page 493
References 470 ......Page 494
Author Index 483 ......Page 507
Subject Index 485 ......Page 509
cover......Page 1
